Stochastic Optimal Control Problems and Parabolic Equations in Banach Spaces

Masiero, Federica

doi:10.1137/050632725

Cited by 37 publications

(56 citation statements)

References 15 publications

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“…In particular it is trivially true in the special case E = H just by taking Ξ 0 = Ξ, since G is assumed to be a linear bounded operator from Ξ to H. The following is proved in [17,Theorem 3.17…”

Section: Then We Consider Again Equation (42)mentioning

confidence: 94%

“…, and apply again [17] (see Proposition 4.2 there) and [12] (see Proposition 5.2 there) to obtain that for all α > 0 the map…”

Section: Then We Consider Again Equation (42)mentioning

confidence: 99%

“…Finally to prove (3.3) we notice that (see the discussion in [17]) the process W A is a Gaussian random variable with values in C([0, T ], E). Therefore by the polynomial growth of F we get…”

Section: And (34) Follows From (32)mentioning

confidence: 99%

“…[21], [9] or, in an infinite dimensional framework, [12], [17]. But up to our best knowledge, there exists no work in which BSDE techniques are applied to optimal control problems with ergodic cost functionals that is functionals depending only on the asymptotic behavior of the state (see e.g.…”

Section: Introductionmentioning

confidence: 99%

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Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces

Fuhrman¹,

Hu²,

Tessitore³

2009

SIAM J. Control Optim.

118

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In this paper we introduce a new kind of Backward Stochastic Differential Equations, called ergodic BSDEs, which arise naturally in the study of optimal ergodic control. We study the existence, uniqueness and regularity of solution to ergodic BSDEs. Then we apply these results to the optimal ergodic control of a Banach valued stochastic state equation. We also establish the link between the ergodic BSDEs and the associated Hamilton-JacobiBellman equation. Applications are given to ergodic control of stochastic partial differential equations.

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Section: Then We Consider Again Equation (42)mentioning

confidence: 94%

“…, and apply again [17] (see Proposition 4.2 there) and [12] (see Proposition 5.2 there) to obtain that for all α > 0 the map…”

Section: Then We Consider Again Equation (42)mentioning

confidence: 99%

Section: And (34) Follows From (32)mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces

Fuhrman¹,

Hu²,

Tessitore³

2009

SIAM J. Control Optim.

118

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“…Using regularizing properties of stochastic convolutions, Masiero (2008) proved existence of mild solutions of a certain class of autonomous HJB equations on the space of continuous functions C(O). Under additional, somewhat restrictive conditions on the nonlinear coefficient F, particularly a dissipative-type condition and a very specific form of dependence with respect to the control variable, Masiero also solved the control problem using backward SDEs but with no use of Malliavin calculus.…”

Section: Dx(t) + A(t)x(t) Dt = F(t X(t) U(t)) Dt + G(t) Dw(t)mentioning

confidence: 99%

Backward Ornstein-Uhlenbeck Transition Operators and Mild Solutions of Non-Autonomous Hamilton-Jacobi Equations in Banach Spaces

Serrano¹

2014

JMR

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In this paper we revisit the mild-solution approach to second-order semi-linear PDEs of Hamilton-Jacobi type in infinite-dimensional spaces. We show that a well-known result on existence of mild solutions in Hilbert spaces can be easily extended to non-autonomous Hamilton-Jacobi equations in Banach spaces. The main tool is the regularizing property of Ornstein-Uhlenbeck transition evolution operators for stochastic Cauchy problems in Banach spaces with time-dependent coefficients.

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HJB equations in infinite dimensions under weak regularizing properties

Masiero

2016

J. Evol. Equ.

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We solve in mild sense Hamilton Jacobi Bellman equations, both in an infinite dimensional Hilbert space and in a Banach space, with lipschitz Hamiltonian and lipschitz continuous final condition, and asking only a weak regularizing property on the transition semigroup of the corresponding state equation. The results are applied to solve stochastic optimal control problems; the models we can treat include a controlled stochastic heat equation in space dimension one and with control and noise on a subdomain.

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Stochastic Optimal Control Problems and Parabolic Equations in Banach Spaces

Cited by 37 publications

References 15 publications

Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces

Ergodic BSDEs and Optimal Ergodic Control in Banach Spaces

Backward Ornstein-Uhlenbeck Transition Operators and Mild Solutions of Non-Autonomous Hamilton-Jacobi Equations in Banach Spaces

HJB equations in infinite dimensions under weak regularizing properties

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